Multiple modular symmetries as the origin of flavor

We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries S-4(A), S-4(B), and S-4(C), associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bitriplet scalars to the diagonal S-4 subgroup. The low energy effective theory involves the three independent moduli fields tau(A), tau(B), and tau(C), which preserve the residual modular subgroups Z(3)(A), Z(2)(B), and Z(3)(C), in their respective sectors, leading to trimaximal TM1 lepton mixing, consistent with current data, without flavons.